Sample allocation for efficient model-based small area estimation

Transcription

1 Catalogue no X ISSN Survey Methoology Sample allocation for efficient moel-base small area estimation by Mauno Keto an Erkki Pahkinen Release ate: June, 017

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3 Survey Methoology, June Vol. 43, No. 1, pp Statistics Canaa, Catalogue No X Sample allocation for efficient moel-base small area estimation Mauno Keto an Erkki Pahkinen 1 Abstract We present research results on sample allocations for efficient moel-base small area estimation in cases where the areas of interest coincie with the strata. Although moel-assiste an moel-base estimation methos are common in the prouction of small area statistics, utilization of the unerlying moel an estimation metho are rarely inclue in the sample area allocation scheme. Therefore, we have evelope a new moel-base allocation name g1-allocation. For comparison, one recently evelope moel-assiste allocation is presente. These two allocations are base on an ajuste measure of homogeneity which is compute using an auxiliary variable an is an approximation of the intra-class correlation within areas. Five moel-free area allocation solutions presente in the past are selecte from the literature as reference allocations. Equal an proportional allocations nee the number of areas an area-specific numbers of basic statistical units. The Neyman, Bankier an NLP (Non-Linear Programming) allocation nee values for the stuy variable concerning area level parameters such as stanar eviation, coefficient of variation or totals. In general, allocation methos can be classifie accoring to the optimization criteria an use of auxiliary ata. Statistical properties of the various methos are assesse through sample simulation experiments using real population register ata. It can be conclue from simulation results that inclusion of the moel an estimation metho into the allocation metho improves estimation results. Key Wors: Optimal area sample size; Criteria; Auxiliary information; Measure of homogeneity. 1 Introuction In this paper we present a new moel-base allocation metho in stratifie sampling where the areas of interest coincie with the strata. Our stuy is focuse on the components of an efficient area allocation. A clear starting point for the allocation process is reache if the areas of interest are efine as early as in the esign phase of the research an if it is also known how large a sample is allowe in consieration of the isposable resources (time, buget etc.). The choice of the allocation metho epens on various factors such as the selecte moel, estimation metho, available pre-information of the population an the optimization criteria set only on area or population level, or on both levels simultaneously. We have selecte six existing allocation methos an evelope a new one which we call a moel-base allocation. The general properties of these methos are examine in Section an Section 3. Five of these allocations can be regare as moel-free. Two of them use only number-base information, such as the number of areas an the number of basic units in each area. Three other allocations nee, in aition to number-base information, area level parameter information, such as area totals, stanar eviation or coefficient of variation (CV). Because this information about the stuy variable is not available, a common solution is to replace it with a proper proxy variable. The last of the reference allocations, introuce by Molefe an Clark (MC) (015), is a moel-assiste allocation which is base on a composite estimator an a two-level moel. We have name it MC-allocation. The optimization criteria of the five moel-free allocations iffer from one another. Allocations base only on area-specific numbers can be compute easily, but their choice is reasonable uner limite 1. Mauno Keto, University of Jyväskylä. mauno.j.keto@stuent.jyu.fi; Erkki Pahkinen, epartment of Mathematics an Statistics of University of Jyväskylä. pahkinen@maths.jyu.fi.

4 94 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation circumstances. In each of the parameter-base allocations the optimization criterion is ifferent. It can be set on the level of the population parameter estimate (Neyman allocation) or on area level estimates in average (Bankier allocation). The thir allocation solution, which eviates from the two former ones, is the NLP allocation, in which the tolerances of estimates are set on both population an area level. This article starts from the assumption that if moel-assiste or moel-base estimation is use in a survey the moel an estimation metho must be taken into account when the allocation of the sample into areas is esigne. This was use as a starting point when the new moel-base g1 allocation, presente in Section, was erive. Also, one of the reference allocations, moel-assiste allocation, is base on a given moel. The comparison of performances of ifferent allocation methos in real situations has been implemente by using simulation experiments an is presente in Section 4. An official Finnish register of block apartments for sale serves as the population. The structure of the register is introuce in Section 4.1. An auxiliary variable has been use in place of the stuy variable when computing the area sample sizes for each allocation except equal an proportional allocation. The comparison emonstrates clearly that these allocations lea to ifferent sample istributions. The same kin of variety also concerns their performances. We have applie moel-base EBLUP (Empirical Best Linear Unbiase Preictor) estimation on the allocations when estimating the area totals of the stuy variable. For measuring an comparing the performances of allocations, a relative root mean square error RRMSE% an absolute relative bias ARB% were use. In Section 5 empirical simulation results are iscusse as concluing remarks. They support the allocation solution in which not only auxiliary information, but also the moel an estimation metho shoul be etermine as early as in the esign phase of a survey. A goo example is the g1 allocation presente in Section.. The most accurate area estimates of area totals were obtaine by using this metho. Allocations which utilize the moel.1 Choosing the moel Pfeffermann (013) presents a wie variety of moels an methos for small area estimation. Our moel is one of this assortment, a unit-level mixe moel y x β v e ; k 1,, N ; 1,,, (.1) k k k where v s are ranom area effects with mean zero an variance an e s are ranom effects with v k mean zero an variance. Furthermore, E e yk x β an k V yk σ σ (total variance). Matrix v e V is the variance-covariance matrix of the stuy variable y. This moel can be use when unit-level values are available for the auxiliary variables x. We use one auxiliary variable in our stuy. Two important measures are neee in eveloping one of these types of allocations. The first one is a common intra-area correlation an the secon one is the ratio between variance components. They are efine as follows: Statistics Canaa, Catalogue No X

5 Survey Methoology, June v v e σ σ σ an σ σ 1 1. (.) Before estimating area parameters, the variance components, regression coefficients an area effects must be estimate from the sample ata. The BLUE estimator (Best Linear Unbiase Estimator) of β, note β, is obtaine accoring to the theory of the general linear moel, an it is replace with its EBLUP estimate β ˆ. The EBLUP estimate (preicte value) for the area total Y of the stuy variable is the sum of the observe y values an preicte y values for units outsie the sample: e v x βˆ (.3) Yˆ y yˆ y N n vˆ.,eblup k k k k ks ks ks ks We use the Prasa-Rao approximation (See Rao 003) of MSE (Mean Square Error) for finite populations: Yˆ g σˆ σˆ g σˆ σˆ g σˆ σˆ g σˆ σˆ mse,,,,, (.4) where the four components g,, 1, Eblup 1 v e v e 3 v e 4 v e g 3 ˆ g σˆ, σˆ N n 1 γ σˆ, 1 v e v g an g are efine as follows: 4 g σ, σ N n, 1 ˆ ˆ ˆ 1 x x X V X x ˆ x v e, ˆ ˆ ˆ ˆ ˆ ˆ 3 v e v e e v g σ σ N n n σ σ n σ V σ σˆ V σˆ σˆ σˆ Cov σˆ, σˆ, 4 v e e v e v, g σˆ σˆ N n σ ˆ. 4 v e e (.5) The area sample sizes n epen on the sample an are not fixe. The component g contains the areaspecific ratio γˆ σˆ σˆ σˆ n. Accoring to Nissinen (009, page 53), the g component (later 1 v v e 1 simply g 1) contributes generally over 90% of the estimate MSE. This component represents uncertainty as regars the variation between the areas. Of course this variation must be strong enough so that such a high proportion for g 1 exists. Unfortunately, the iea of an analytical solution, which means minimizing the sum of MSE s over areas subject to n n, is ifficult an laborious to accomplish because components of the MSE 1 approximation (.5) inclue sample information which is unknown, an some components contain complex matrix an variance-covariance operations. We have examine this allocation problem for the first time in an experimental stuy (Keto an Pahkinen 009). Now we have evelope an allocation base only on the component g 1 an auxiliary variable x. The reasoning for this solution is that because x an y are correlate, the between-area variation in x is transferre to y. Statistics Canaa, Catalogue No X

6 96 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation. Moel-base g1allocation The g1 allocation utilizes the auxiliary variable x an the ajuste homogeneity coefficient (Keto an Pahkinen 014). This coefficient is an approximation of an intra-class correlation (ICC) known of cluster sampling. We regar one area as one cluster. First, simple ANOVA between areas is carrie out, an then the ajuste homogeneity measure of variation between the areas can be compute: R 1 R x 1 MSW S, (.6) ax where R x is the coefficient of etermination from regression analysis, MSW (Mean Square within) is the mean SS (Sum of Squares) of areas an S is the variance of the auxiliary variable x. x Because MSE of the area total is complex, we use only the component g 1, which appears in (.4) an (.5), for the reason we have given in Section.1. We search for the minimum for the sum of g 1 s over areas: x g σ, σ v e N n n σ σ (.7) e v 1 1 subject to n n. 1 We use Lagrange s multiplier metho to fin the solution. Therefore, we efine the function F of sample sizes n n, n,, n 1 an : 1 Fn, g1 σ, σ N n n σ 1 σ n n. v e e v (.8) We set the erivative of F with respect to the area sample size n to zero an solve for n. The expression g1 for area sample size n is as follows: N δn δ N nn N n δ N δ N n, g (.9) where the ratio an the intra-area correlation are efine in (.). The only unknown member in (.9) is the intra-area correlation. Therefore we substitute the known homogeneity measure (.6) of the auxiliary variable x for. Thus the final expression for computing area sample sizes is n N 1 R 1 N n N N n 1 R 1. g1 ax ax (.10) It is easy to prove that g1 n n. The compute sample sizes are roune to the nearest integer. 1 Sometimes compromises must be mae. It can be conclue by the examination of (.10) that the sample size increases when the size of area N increases, but not proportionally. Uner certain circumstances, such N can lea to negative as low homogeneity coefficient, low overall sample size n or small size of area, area sample size n In this situation the negative value is change to zero. A special case occurs if the g1. total variation is only between areas causing value one to the measure of homogeneity (.6), an (.10) is reuce to proportional allocation. Statistics Canaa, Catalogue No X

7 Survey Methoology, June Moel-assiste MC-allocation Molefe an Clark (015) have use the following composite estimator for estimating the mean of the stuy variable y for area : y ˆ C y 1 β X. (.11) r This estimator is a combination of two estimators: the synthetic estimator Yˆ ˆ syn βx, where ˆβ is the estimate regression coefficient an X is the area population means of auxiliary variables x, an a irect estimator y y βˆ x X, where y an x are the area sample means of y an x. We use r one auxiliary variable in our stuy. The coefficients are set with the intent to minimize the MSE of the estimator (.11). The approximate esign-base MSE of the estimator uner certain conitions an assumptions is given by the expression C syn MSE y ; Y 1 v B, p (.1) where v syn is the sampling variance of the synthetic estimator Y ˆ syn an B β X Y is the bias U when Y ˆ syn is use to estimate Y, with β enoting the approximate esign-base expectation of β ˆ. U The population contains N units an strata efine by areas, an stratifie sampling is use. A ranom sample SRSWOR (Simple Ranom Sampling without Replacement) of n units is selecte from stratum 1,, containing N units. The relative size of area is P N N. A two-level linear moel conitional on the values of x is assume, with uncorrelate stratum ranom effects u an ranom effects : i y βx u i i i E u E 0 i, (.13) V u u V i e where i refers to all units in stratum. This moel implies that V y i u e for all population units an cov y, y i j equals for units i j in the same stratum an zero for units from ifferent strata, where. A simplifying assumption that are equal for all strata is efine. u u e After making some other simplifying assumptions an solving the optimal weight in (.1), the final approximate optimum anticipate MSE or approximate moel assiste mean square error is obtaine of (.1): 1 C AMSE E MSE y opt ; Y 1 1 n 1. p (.14) Next the criterion F using anticipate MSE s of the small area mean an overall mean estimators for moelassiste allocation is efine an evelope into the final approximative form: 1 q var ˆ q F N AMSE GN E Y p r q 1 q 1 N σ 1 1 n 1 GN P n 1. (.15) 1 1 Statistics Canaa, Catalogue No X

8 98 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation Optimal sample sizes for the areas are obtaine by minimizing (.15) subject to n n. Expression (.15) follows the iea of Longfor (006). The weight N reflects the inferential priority (importance) for area, with 0 q, an q q N N. The quantity G is a relative priority coefficient on the 1 population level. Ignoring the goal of estimating the population mean correspons to G 0, an the attention is then only focuse on area level estimation. On the other han, the larger the value of G, the more the secon component in (.15) ominates an the more the area level estimation is ignore. We assume first that the population estimation has no priority G 0 an the unit survey cost are fixe. In this case minimization of (.15) with respect of n has a unique solution q n n N 1 N 1. q q,opt q 1 q N N 1 1 (.16) The formula (.16) contains two unknown parameters, the intra-class correlation an the area-specific variance. We replace with an ajuste homogeneity coefficient of the auxiliary variable x. This coefficient is an approximation of the ICC (Intra-Class Correlation) (Section.). Parameter is replace with the variance of x in area. The reason for both replacements is that y is correlate with x. If also the population estimation has a priority G 0 then (.16) oes not apply an F must be minimize numerically by using, for example, the NLP metho, as we have one (Excel Solver, NLP option). Table.1 Summary of moel-base an moel-assiste allocations Metho Moel-base g 1 Moel-assiste MCG0 MCG50 Computing sample size n for area N n N N n 1 1 R 1 g ax n, N 1 R 1 ax where R is the ajuste homogeneity measure of auxiliary variable x. ax n n N 1 N 1 q q,opt q 1 q N N 1 1 Minimization of 1 q q F N σ n GN P n with respect of n. Parameter is replace with R an with ax S x. Optimality level Area Jointly area an population 3 Some moel-free area allocations The aim of this section is to list the five previously presente allocation methos in orer to use them later as references. epening on which kin of auxiliary information each one uses, they are ivie into two groups: number-base an parameter-base allocations. Statistics Canaa, Catalogue No X

9 Survey Methoology, June Number-base allocations Two basic allocation solutions commonly use go uner the names equal allocation an proportional allocation. Neither of these allocations contains any specific criterion on the area or population level. Their implementation requires only information on the number of strata an the numbers of units stratum. In the equal area allocation the sample size n is simply a quotient n Equ N in each n. (3.1) It is recommene to choose the total sample size n so that the quotient is a whole number. This allocation metho oes not take ifferences between the areas into account in any way, which results in inaccurate area estimates. A natural lower limit of the sample size is min n. Proportional allocation is a frequently use basic metho. Area sample sizes are solve from Pro. n n N N (3.) If the sizes of the areas vary strongly, it can lea to situations where the allocate sample size Pro n for one or more areas. This is an obstacle in calculating irect esign-base estimates of stanar errors. One solution is to apply the combine allocation propose by Costa, Satorra an Ventura (004). The iea is a weighte solution between the equal an proportional allocation epening on the situation. The combine area sample size is Com Pro Equ n kn 1 k n (3.3) for a specifie constant k0 k 1. A minor problem is present if for some areas n N. A moifie solution exists for this case. 3. Parameter-base allocations These allocations use area-level information of the stuy variable y an in some cases of the auxiliary variable x correlate with y. The values of x are available for all population units. In practice the unknown y is replace with a proper proxy variable y such as a stuy variable obtaine from an earlier research of the same subject, or the values of y are generate with a suitable moel evelope of a small pre-sample. Also x can be substitute for y. Allocation criteria can be set on population level, only on area level or on combine population an area level. The Neyman allocation aims at reaching an optimal accuracy concerning population parameters Sy (Tschuprow 193). The stanar eviation of the stuy variable y or some proxy variable an the number of units in each area must be known. Allocation favors large areas with strong variation. The Bankier or power allocation (1988) is base on a criterion set on the area level. Area CV values of q y are weighte by area total transformations X which contain a tuning constant q. In practice y or x must be use in place of y. Allocation favors mainly large areas with high CV. Chouhry, Rao an Hiiroglou (01) present the NLP allocation metho for irect estimation. This metho uses non-linear programming to fin a solution. Criteria for the allocation are efine by setting Statistics Canaa, Catalogue No X

10 100 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation upper limits for CV values of the stuy variable y in each area an in the population. In practice y or x replaces y. The program searches the minimum sample size n n satisfying these conitions. The SAS (Statistical Analysis System) proceure NLP with Newton-Raphson option was use to fin the solution. The allocation favors areas with high CV regarless of the area size N. A summary of the moel-free allocations an the formulas for calculating area sample sizes are presente in Table 3.1. Table 3.1 Summary of number-base an parameter-base allocations Allocation Computing area sample size n Optimality level Equal n Equ n Area Proportional Pro n nn N Population Ney Neyman n nn S N S 1, where S is the stanar eviation of y (in practise y or x ) in area. Population Ban q q Bankier n n X CVy CV, X y where X is the area total of x, 1 y S Y an q is a tuning constant. In practise y or x CV Area replace y. NLP NLP n min n st 1 satisfying tolerances CV CV 0 y an Jointly population an area CVy CV. In practise y or x replace y. st 0 Some other parameter-base allocation methos are mentione briefly. For example Longfor (006) introuce inferential priorities P for the strata an G for the population an use those constraints for allocation. Another solution is presente by Falorsi an Righi (008). This solution oes not contain a irect imposition of quotas, but tries to solve the comprehensive collection of ata by using a multi-stage sampling esign, so that the area estimation can be implemente effectively. 4 Comparison of performances of allocations In this section we stuy the performances of the allocation methos introuce in Sections an 3. The estimate parameters are area an population totals of the stuy variable y. The overall sample size n 11. Section 4.1 inclues the escription of the research ata. Simulation experiments an comparisons of allocations are presente in Section 4.3. Statistics Canaa, Catalogue No X

11 Survey Methoology, June Empirical ata Our research ata is obtaine from a national Finnish register of block apartments for sale. This register is maintaine by a private company, Alma Meiapartners Lt, whose customers are real estate agencies. They save all the necessary information of the apartments into this register as soon as they receive an assignment from the owners. The population we have use consists of 9,815 block apartments (these serve as sampling units) for sale selecte from the register. They represent 14 Finnish istricts, mainly towns, in spring 011. The sizes of the smallest an largest area were 11 an 1,333, respectively. The stuy variable y measures the apartment price (1,000 ) an the auxiliary variable x measures the size (m ). Area sizes N, population summary statistics (totals, means, stanar eviations an CVs) for y an x, as well as correlations between x an y, are given in Table 4.1. The characteristics of the areas have a wie range. The most iverging area is Helsinki. Table 4.1 Population summary statistics Area Stuy variable y Auxiliary variable x Correlation Label N Y Y S y CV y X X S x CV x r yx Porvoo town 11 5, , Pirkkala istrict , , South Savo county , , Jyväskylä town , , Lappi county 555 6, , South-East Finlan , , Helsinki (capital) , , West coast istrict , , Tracksie istrict , , Kuopio istrict , , Turku istrict , , Oulu istrict 1,07 133, , Metropol area 1,100 63, , Lahti-Tampere istr. 1,333 6, , Population 9,815,019, , The ajuste measure of homogeneity of the auxiliary variable x is R 0.31 inicating quite strong ax variability between the areas. 4. Allocations In general, the overall sample size epens on the available time an financial resources in the research project. This aspect has not been taken into account now, because it is a question of an experimental stuy. Statistics Canaa, Catalogue No X

12 10 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation The value of the sampling ratio was etermine as f % , %. Metho-specific allocations were prouce accoring to the formulas presente in Table.1 an Table 3.1. Some etails have been taken into account. In the Bankier allocation the value of a tuning constant q is 0.5. In the NLP allocation the selecte CV limits (1.58%) for areas an the CV limit (3.75%) for the population lea to the overall sample size 11. We use the Excel Solver proceure with non-linear option for solving the NLP allocation problem. We use a moifie proportional allocation to obtain an area sample size which is at least two. First we allocate one unit for every area an then allocate the rest 98 units by using proportionality. We have substitute x for y in every parameter-base allocation. In the moelassiste allocations the value of q was set to 1, an the quantity G was set to zero an 50. The final sample sizes in each allocation are presente in Table 4.. The variation of sample sizes on area level is very strong between the allocations. Table 4. Area sample sizes by allocation Area Label Moelbase 1 N g Composite estim. Moel-assiste MCG0 Number-base allocations Parameter-base allocations MCG50 EQU PRO Ney _ X Ban _ X NLP _ X Porvoo town Pirkkala istrict South Savo county Jyväskylä town Lappi county South-East Finlan Helsinki (capital) West coast istrict Tracksie istrict Kuopio istrict Turku istrict Oulu istrict 1, Metropol area 1, Lahti-Tampere istrict 1, Total 9, base on the ajuste coefficient of homogeneity (value 0.31) compute of x. 4.3 Comparison of performances of allocations In this section we present the results base on esign-base simulation experiments. For each allocation, 1,500 inepenent stratifie SRSWOR samples were simulate with the SAS program an necessary calculations from the simulate samples were implemente with SPSS (Statistical Package for the Social Sciences) program. We have applie moel-base EBLUP estimation on the samples for each allocation. For comparison of the allocations, we have compute two quality measures: RRMSE % an ARB % for each allocation. Assume that r simulate samples are rawn in each allocation, an let Y be the EBLUP estimate ˆi,EBLUP th of the area total Y in the i sample i 1,, r. Then RRMSE % an ARB % are efine as Statistics Canaa, Catalogue No X

13 Survey Methoology, June r RRMSE % r Yˆ Y Y, i1 i,eblup r ARB % Yˆ Y 1, i1 i,eblup an their means over areas are compute as follows: MRRMSE% 1 RRMSE % an MARB% 1 ARB %. 1 1 th The estimate for the population total in the i simulate sample i 1,, r is the sum of the estimates of the area totals: ˆ Y Y ˆ.,EBLUP RRMSE% for the population total is compute as,eblup i 1 i pop i1 ˆ RRMSE % r Y Y Y, r i,eblup where Y is the true value of the population total, for which ARB% is compute as r ARB % r Yˆ Y 1. pop i1 i,eblup Tables 4.3 an 4.4 contain RRMSE% an ARB% values for areas, their means over areas an population RRMSE%s an ARB%s in each allocation. The evaluation of the results was base on two arguments. One was the mean value of the quality measure on the area level an the other was the value of the quality measure on the population level. Table 4.3 Area an population RRMSE%s by allocation Area N g 1 MCG0 MCG50 EQU PRO Ney_X Ban _ X NLP _ X Porvoo town Pirkkala istrict South Savo county Jyväskylä town Lappi county South-East Finlan Helsinki (capital) West coast istrict Tracksie istrict Kuopio istrict Turku istrict Oulu istrict 1, Metropol area 1, Lahti-Tampere istr. 1, Mean over areas (%) Population value (%) The lowest RRMSE% mean over the areas (15.65%) was obtaine in the g1 allocation evelope in this stuy. Helsinki was an exception on area level because its RRMSE% value was clearly higher compare Statistics Canaa, Catalogue No X

14 104 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation with moel-assiste an parameter-base allocations. Also equal an proportional allocations performe well on area level, with means 16.48% an 16.5%. The highest means were obtaine in the moel-assiste MC-allocations. On the population level, the lowest value for the quality measure was obtaine in the moelassiste MCG50-allocation (5.88%) an the secon lowest value in the Bankier allocation (5.89%), but in general, ifferences between the allocations on this level were small. Table 4.4 Area an population ARB%s by allocation Area N g 1 MCG0 MCG50 EQU PRO Ney _ X Ban _ X NLP _ X Porvoo town Pirkkala istrict South Savo county Jyväskylä town Lappi county South-East Finlan Helsinki (capital) West coast istrict Tracksie istrict Kuopio istrict Turku istrict Oulu istrict 1, Metropol area 1, Lahti-Tampere istr. 1, Mean over areas (%) Population value (%) The g1 allocation was the only allocation with absolute relative bias less than 10% on each area, an it ha a practically zero bias on the population level. Also the equal an proportional allocations ha low biases on both levels, but the moel-assiste an parameter-base allocations ha a clearly poorer performance. An interesting etail in the g1 allocation is that the accuracy of area estimates is fairly goo an the relative bias is low also for the case of two areas with zero sample size. A common characteristic for these areas is that the means of variables y an x are close to corresponing population means. In any case, it is essential that the moel-base estimation can prouce reliable estimates for areas, which are not represente in the ranom sample. 5 Concluing remarks This research was focuse on seven ifferent allocation solutions which were categorize into three groups accoring to the auxiliary ata neee in their implementation. The least amount of auxiliary information is neee in equal an proportional allocation which are base on the number of areas an the number of statistical units in each area. The Neyman, Bankier an NLP allocations are base on pre-set optimization criteria, an application of these methos presumes area-specific parameter information such Statistics Canaa, Catalogue No X

15 Survey Methoology, June as the stanar eviation or CV of the stuy variable, an in the Bankier allocation the area totals of at least one auxiliary variable must be known. Because the stuy variable is unknown, it must be replace with a suitable proxy or auxiliary variable to enable the use of these three methos. A common feature of the number-base an parameter-base allocations is that they are not base on any moel, whereas the other three allocations utilize the unerlying moel, in aition to number-base information. On the basis of the empirical results, the performance of the moel-base g1 allocation can be regare as the best compare with the other allocations teste in this research. Also equal an proportional allocations reache goo results, but the moel-assiste allocations an the parameter-base allocations ha clearly weaker performances. The last three allocations are evelope originally for irect esign-base estimation, an their results can be unerstoo from that point of view. Compare with g1 allocation, the MC-allocations are base on a ifferent moel an this fact seems to affect their results. One of the characteristics of the g1 allocation is that when the sampling esign is constructe, also the moel an estimation metho are fixe, meaning that they are regare as given preliminary information. This allocation, which is base on a unit-level linear mixe moel an EBLUP estimation metho, nees only the homogeneity coefficient between areas which is compute by using the values of the auxiliary variable. In this respect, the g1 allocation iffers from the other allocations use in the comparison. Also the starting point for choosing the final estimation metho is ifferent, because this allocation is focuse on moel-base estimation, not on irect esign-base estimation using sampling weights. The choice of the moel-base estimation is justifie also for the reason that it is commonly use in small area estimation. On the other han, the g1 allocation enables the use of small sample sizes, because information can be borrowe between areas when the moel is applie. This can be significant in quick surveys or stuies carrie out by market research organizations, when a single measurement is expensive. However, it is important to examine the characteristics of the areas an especially the small areas, before the final sample sizes are etermine. As a recommenation, it woul be justifie to start a wier research to fin out what avantages an isavantages are encountere if the applicable computing technique for proucing area statistics is ecie as early as in the esign of the research plan. Acknowlegements The authors thank the Eitor, Associate Eitor an two referees as well as Professor Risto Lehtonen for constructive comments an suggestions. References Bankier, M.. (1988). Power allocations: etermining sample sizes for subnational areas. The American Statistician, 4, Chouhry, G.H., Rao, J.N.K. an Hiiroglou, M.A. (01). On sample allocation for efficient omain estimation. Survey Methoology, 38, 1, 3-9. Paper available at x/01001/article/1168-eng.pf. Statistics Canaa, Catalogue No X

16 106 Keto an Pahkinen: Sample allocation for efficient moel-base small area estimation Costa, A., Satorra, A. an Ventura, E. (004). Improving both omain an total area estimation by composition. SORT, 8(1), Falorsi, P.., an Righi, P. (008). A balance sampling approach for multi-way stratification for small area estimation. Survey Methoology, 34,, Paper available at Keto, M., an Pahkinen, E. (009). On sample allocation for effective EBLUP estimation of small area totals Experimental Allocation. In Survey Sampling Methos in Economic an Social Research, (Es., J. Wywial an W. Gamrot), 010. Katowice: Katowice University of Economics. Keto, M., an Pahkinen, E. (014). On sample allocation for efficient small area estimation. Book of Abstracts. SAE 014, Polan: Poznan University of Economics, page 50. Longfor, N.T. (006). Sample size calculation for small-area estimation. Survey Methoology, 3, 1, Paper available at Molefe, W.B., an Clark, R.G. (015). Moel-assiste optimal allocation for planne omains using composite estimation. Survey Methoology, 41,, Paper available at Nissinen, K. (009). Small Area Estimation with Linear Mixe Moels from Unit-Level Panel an Rotating Panel ata. Ph.. thesis, University of Jyväskylä, epartment of Mathematics an Statistics, Report 117, Pfefferman,. (013). New important evelopments in small area estimation. Statistical Science, 8, Rao, J.N.K. (003). Small Area Estimation. Hobogen, New Jersey: John Wiley & Sons, Inc. Tschuprow, A.A. (193). On the mathematical expectation of the moments of frequency istributions in the case of correlate observations. Metron, Vol., 3, ; 4, Statistics Canaa, Catalogue No X